Blumenthal, Leonard M. Formerly, Department of Mathematics, University of Missouri, Columbia, Missouri.
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A member of the class of curves that are intersections of a plane with a cone of revolution. The ellipse is obtained when the plane cuts all the elements of one nappe, and does not go through the apex. In the illustration, denote the distance between two points, F, F′ of a plane by 2c, c > 0, and let 2a be a constant, with a > c. The ellipse with foci F and F′ and major axis 2a is the locus of points P of the plane such that PF + PF′ = 2a, where PF denotes the distance of P and F. This suggests the following construction of an ellipse. Put pins at F and F′, and slip over them a loop of thread of length 2a + 2c, pulling the thread taut with a pencil. If the pencil is moved, keeping the thread taut, its point traces an ellipse. Another way to construct an ellipse is to drill a hole in a stick (at any point other than the midpoint) and move the stick so that its ends slide along two mutually perpendicular lines. The point of a pencil inserted in the hole will trace an ellipse. Limiting forms of the ellipse are (1) a circle, as the two foci approach coincidence, and (2) the segment FF′, as c approaches a. If a circle is projected orthogonally on a plane not parallel to the plane of the circle, an ellipse is obtained, and every ellipse may be so obtained. Lines joining the foci to a point P of an ellipse make equal angles with the tangent to the ellipse at P, and consequently light or sound that emanates from one focus is reflected to the other focus. This property is used in construction of “whispering galleries.” See also: Conic section
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